AMC 12 2005b Problem 8

For how many values of $a$ is it true that the line $y = x + a$ passes through the vertex of the parabola $y = x^2 + a^2$ ?

$\mathrm{(A)}\ 0 \mathrm{(B)}\ 1 \mathrm{(C)}\ 2 \mathrm{(D)}\ 10 \mathrm{(E)}\ \text{infinitely many}$